The present invention relates to a method of calculating the resistance value of an impedance of a conductive loop formed by a fault in a monitored section of an electric power supply line through which an a.c. current is flowing, using instantaneous values of the measured current and the measured voltage sampled in an interval of time.
European Patent No. 0 284,546 describes a method of calculating a resistance. In this method, instantaneous values of the current and the voltage are sampled and weighted in a digital filter unit composed of multiple linear-phase non-recursive digital FIR filters of a first type and a second type. The weighted instantaneous values are used in first-order differential equations. These differential equations are then used to form a resistance value. European Patent No. 0 284,546 further describes that with a sampling period of 1 ms and for alternating current with a period of oscillation of 20 ms, a total of 13 instantaneous values of the current and/or the corresponding voltage must be weighted for a rapid and sufficiently accurate determination of the resistance value (page 4, lines 7 through 11, and page 9, lines 45 through 47). This number of instantaneous values is obtained from a selected counter degree of n=8 and from an interval of approximately xc2xc the system period (5 ms) which is to be maintained in sampling linearly independent instantaneous values. Thus, sampling the instantaneous values requires 13 ms.
An object of the present invention is to reduce the time required to form the resistance value the type in comparison with the conventional method, without thereby becoming much more inaccurate.
This object is achieved according to the present invention by providing a method including the steps of:
detecting instantaneous values of the current and voltage sampled during half a period of the a.c. current,
calculating instantaneous values of the power from the instantaneous values detected for the current and voltage, and determining from them a value proportional to the effective power by integration,
squaring the instantaneous values of the current thus determined and determining an integral of the squared instantaneous value of the current, and
generating the resistance value from the value proportional to the effective power and the value of the integral of the squared instantaneous values of the current by forming a quotient.
Unexamined German Patent Application No. 25 51 472 describes a device for testing resistance welding equipment. A resistance value is obtained by calculating the quotient of an intermediate measured quantity proportional to the effective power and another intermediate measured quantity proportional to the effective current value. However, this value gives the resistance value of a welding cable of the resistance welding equipment. The current through the welding cable and the voltage drop on this cable are detected and processed further as analog (sinusoidal) values. After integration, the current is sent first to a squaring element with a downstream integrator and also, together with the voltage drop, sent to a multiplier with a downstream integrator circuit. At the output of the integrator circuit, an intermediate measured quantity is obtained, while the additional intermediate measured quantity is obtained at the output of the integrator. Because of the processing of analog values, this conventional device requires a measurement time of at least one period of the current through the welding cable to form the resistance value.
An important advantage of the method according to the present invention is that for mathematical reasons, to determine the resistance values, it is sufficient to determine the instantaneous values during only half a period of the a.c. oscillation, i.e., with an oscillation period of 20 ms, only 10 ms are needed to detect the instantaneous values. The method according to the present invention makes use of the property of the alternating current whereby the current and the corresponding voltage can be described with the following sine functions:
I(t)={square root over (2)}xc2x7I sin(xcfx89t+"psgr")
U(t)={square root over (2)}xc2x7U sin(xcfx89t).
where I and U here are the effective current and voltage values, xcfx89 is the angular frequency of the system oscillation, and xcfx86 is the phase difference between I(t) and U(t).
The product of the current times voltage yields the following equation:
I(t)xc2x7U(t)={square root over (2)}xc2x7I sin(xcfx89t+t)xc2x7{square root over (2)}U sin("psgr"t)
After converting, this yields
I(t)xc2x7U(t)=UI cos"psgr"xe2x88x92UI cos(2xcfx89t +"psgr").
The right side of this equation has a cosine with twice the system oscillation frequency 2xcfx89. The above-mentioned advantage derives from the fact that in integration of the product I(t)xc2x7U(t) over a period T2xcfx89, which corresponds to half a system oscillation period, the values of the cosine with twice the system oscillation frequency add up to zero. The following value which is proportional to the effective power P=UIcosxcfx86 is obtained:                                           ∫                          T                              2                ⁢                ω                                                    xe2x80x83                                ⁢                                                    U                ⁡                                  (                  t                  )                                            ·                              I                ⁡                                  (                  t                  )                                                      ⁢                          xe2x80x83                        ⁢                          ⅆ              t                                      =                              π            ω                    ⁢          U          ⁢                      xe2x80x83                    ⁢          I          ⁢                      xe2x80x83                    ⁢          cos          ⁢                      xe2x80x83                    ⁢          ϕ                                    (        1        )            
On the other hand, effective power P can also be calculated with the following equation in a known way:
P=I2R,
where R is the resistance value.
The square of the instantaneous values of the current is needed for the calculation. To do so, integration of squared instantaneous current values over the period T2xcfx89is calculated in the following equation:
                                                                                          ∫                                      T                                          2                      ⁢                      ω                                                                            xe2x80x83                                                  ⁢                                                                            I                      2                                        ⁡                                          (                      t                      )                                                        ⁢                                      xe2x80x83                                    ⁢                                      ⅆ                    t                                                              =                              2                ⁢                                  I                  2                                ⁢                                                      ∫                                          T                                              2                        ⁢                        ω                                                                                    xe2x80x83                                                        ⁢                                                                                    sin                        2                                            ⁡                                              (                                                  ωt                          +                                                      xe2x80x83                                                    ⁢                          ϕ                                                )                                                              ⁢                                          xe2x80x83                                        ⁢                                          ⅆ                      t                                                                                                                                              =                                                x                  ω                                ⁢                                  I                  2                                                                                        (        2        )            
It follows that resistance value R can be determined from the values of equations (1) and (2):                                                         R              =                                                                    ∫                                          T                                              2                        ⁢                        ω                                                                                    xe2x80x83                                                        ⁢                                                                                    U                        ⁡                                                  (                          t                          )                                                                    ·                                              I                        ⁡                                                  (                          t                          )                                                                                      ⁢                                          xe2x80x83                                        ⁢                                          ⅆ                      t                                                                                                            ∫                                          T                                              2                        ⁢                        ω                                                                                    xe2x80x83                                                        ⁢                                                                                    I                        2                                            ⁡                                              (                        t                        )                                                              ⁢                                          xe2x80x83                                        ⁢                                          ⅆ                      t                                                                                                                                              =                                                U                  ⁢                                      xe2x80x83                                    ⁢                  I                  ⁢                                      xe2x80x83                                    ⁢                  cos                  ⁢                                      xe2x80x83                                    ⁢                  ϕ                                                  I                  2                                                                                        (        3        )            